Square‐integrable solutions and Weyl functions for singular canonical systems

Boundary value problems for singular canonical systems of differential equations of the form\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ Jf^{\prime }(t)-H(t) f(t)=\lambda \Delta (t) f(t),\qquad t\in \imath ,\quad \lambda \in {\mathbb C}, $$ \end{document} are studied in the associated Hilbert space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2_\Delta (\imath )$\end{document} . With the help of a monotonicity principle for matrix functions their square‐integrable solutions are specified. This yields a direct treatment of defect numbers of the minimal relation and simultaneously makes it possible to assign certain boundary values to the elements of the maximal relation induced by the system of differential equations in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2_\Delta (\imath )$\end{document} . The investigation of boundary value problems for these systems and their spectral theory can be carried out by means of abstract boundary triplet techniques. This paper makes explicit the construction and the properties of boundary triplets and Weyl functions for singular canonical systems. Furthermore, the Weyl functions are shown to have a property similar to that of the classical Titchmarsh‐Weyl coefficients for singular Sturm‐Liouville operators: they single out the square‐integrable solutions of the homogeneous systems of canonical differential equations. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim